Spherical homogeneous spaces of minimal rank

Let $G$ be a complex connected reductive algebraic group and $G/B$ denote the flag variety of $G$. A $G$-homogeneous space $G/H$ is said to be {\it spherical} if $H$ acts on $G/B$ with finitely many orbits. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group $G$ (viewed as a $G\times G$-homogeneous space) has particularly nice proterties. Namely, the pair $(G,H)$ is called a {\it spherical pair of minimal rank} if there exists $x$ in $G/B$ such that the orbit $H.x$ of $x$ by $H$ is open in $G/B$ and the stabilizer $H_x$ of $x$ in $H$ contains a maximal torus of $H$. In this article, we study and classify the spherical pairs of minimal rank.Comment: Document produced in 200

Two generalizations of the PRV conjecture

Let G be a complex connected reductive group. The PRV conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for "the same reason" as the PRV ones

INP1 involvement in pollen aperture formation is evolutionarily conserved and may require species-specific partners

Pollen wall exine is usually deposited non-uniformly on the pollen surface, with areas of low exine deposition corresponding to pollen apertures. Little is known about how apertures form, with the novel Arabidopsis INP1 (INAPERTURATE POLLEN1) protein currently being the only identified aperture factor. In developing pollen, INP1 localizes to three plasma membrane domains and underlies formation of three apertures. Although INP1 homologs are found across angiosperms, they lack strong sequence conservation. Thus, it has been unclear whether they also act as aperture factors and whether their sequence divergence contributes to interspecies differences in aperture patterns. To explore the functional conservation of INP1 homologs, we used mutant analysis in maize and tested whether homologs from several other species could function in Arabidopsis. Our data suggest that the INP1 involvement in aperture formation is evolutionarily conserved, despite the significant divergence of INP1 sequences and aperture patterns, but that additional species-specific factors are likely to be required to guide INP1 and to provide information for aperture patterning. To determine the regions in INP1 necessary for its localization and function, we used fragment fusions, domain swaps, and interspecific protein chimeras. We demonstrate that the central portion of the protein is particularly important for mediating the species-specific functionality.Funding was provided to AAD by the US National Science Foundation (MCB-1517511) and to VNSS by the Spanish Ministry of Economy and Competitiveness (CGL2015-70290-P). PL was supported by the China Scholarship Council. SB-MS was supported by the University of Granada, Spain (grant Cei BioTic). We thank the Arabidopsis Biological Resource Center (OSU) and the Maize Genetics Cooperation Stock Center (USDA/ ARS) for seed stocks, Priscila Rodriguez Garcia (OSU) for help with characterizing Arabidopsis–tomato INP1 chimeras, and Jay Hollick (OSU) for advice on all things maize

Distributions on homogeneous spaces and applications

International audienceLet $G$ be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product $\odot_0$ on thecohomology group $H^*(G/P,{\mathbb C})$ of any projective $G$-homogeneousspace $G/P$.Their definition uses the notion of Levi-movability for triples ofSchubert varieties in $G/P$.In this article, we introduce a family of $G$-equivariant subbundlesof the tangent bundle of $G/P$ and the associated filtration of the DeRham complex of $G/P$ viewed as a manifold. As a consequence, one gets a filtration of the ring $H^*(G/P,{\mathbb C})$and proves that $\odot_0$ is the associated graded product.One of the aim, of this more intrinsic construction of $\odot_0$ isthat there is a natural notion of fundamental class$[Y]_{\odot_0}\in(H^*(G/P,{\mathbb C}),\odot_0)$ for any irreducible subvariety $Y$ of $G/P$.Given two Schubert classes $\sigma_u$ and $\sigma_v$ in$H^*(G/P,{\mathbb C})$, we define a subvariety $\Sigma_u^v$ of $G/P$. This variety should play the role of the Richardson variety; moreprecisely, we conjecture that$[\Sigma_u^v]_{\odot_0}=\sigma_u\odot_0\sigma_v$.We give some evidence for this conjecture and prove special cases.Finally, we use the subbundles of $TG/P$ to give a geometriccharacterization of the $G$-homogeneous locus of any Schubertsubvariety of $G/P$

Distributions on homogeneous spaces and applications

International audienceLet $G$ be a complex semisimple algebraic group. In 2006, Belkale-Kumar defined a new product $\odot_0$ on thecohomology group $H^*(G/P,{\mathbb C})$ of any projective $G$-homogeneousspace $G/P$.Their definition uses the notion of Levi-movability for triples ofSchubert varieties in $G/P$.In this article, we introduce a family of $G$-equivariant subbundlesof the tangent bundle of $G/P$ and the associated filtration of the DeRham complex of $G/P$ viewed as a manifold. As a consequence, one gets a filtration of the ring $H^*(G/P,{\mathbb C})$and proves that $\odot_0$ is the associated graded product.One of the aim, of this more intrinsic construction of $\odot_0$ isthat there is a natural notion of fundamental class$[Y]_{\odot_0}\in(H^*(G/P,{\mathbb C}),\odot_0)$ for any irreducible subvariety $Y$ of $G/P$.Given two Schubert classes $\sigma_u$ and $\sigma_v$ in$H^*(G/P,{\mathbb C})$, we define a subvariety $\Sigma_u^v$ of $G/P$. This variety should play the role of the Richardson variety; moreprecisely, we conjecture that$[\Sigma_u^v]_{\odot_0}=\sigma_u\odot_0\sigma_v$.We give some evidence for this conjecture and prove special cases.Finally, we use the subbundles of $TG/P$ to give a geometriccharacterization of the $G$-homogeneous locus of any Schubertsubvariety of $G/P$

Geometric Invariant Theory and Generalized Eigenvalue Problem

Let G be a connected reductive subgroup of a complex semi-simple group ˆ G. We are interested in the set of pairs (ˆν, ν) of dominant characters for ˆ G and G such that Vˆν ⊗Vν contains nonzero G-invariant vectors. This set of pairs (ˆν, ν) generates a convex cone C in a finite dimensional vector space. Using methods of variation of quotient in Geometric Invariant Theory, we obtain a list of linear inequalities which characterize C. This list is a generalization of the list that Belkale and Kumar obtained in the case when ˆ G = G s. Moreover, we prove that this list in no far to be minimal (and really minimal in the case when ˆG = G s). We also give a description of some lower faces of C; if ˆ G = G s these description gives an application of the Belkale-Kumar product ⊙0 on the cohomology group of all the projective G-homogeneous spaces. Some of the results are more general than in the abstract and are obtained in the general context of Geometric Invariant Theory.